4
1. PRELIMINARIES
Here Q is a bounded domain with the Lipschitz boundary, n is the exterior
(with respect to Q) normal to d£l and u is a function of class C (Q) or
even
Hltl{Q).
A linear differential operator of order m is an operator
|e*|m
If m = 2 it is convenient sometimes to write
(1.1.2) A =
^ajkDjDk
+ X X Dj + * ^ =
*kJ
Then the (formally) adjoint operator is
(1.1.3) 'A = £
Dj(ajkDk)
+ £ Z).(^) + a,
where
ajfc
e
C2
and
aJ
e
Cl
and
Here sums are taken over j , k = 1, ... , n . Since
vAu
-ulAv
=
Y,(Dj(vaJkDku
-
uajkDkv)
+
Dj{(-Dkajk
+
aj)uv)),
then applying the formula (1.1.1) we get the following well-known result.
THEOREM
1.1.3
(GREEN'S FORMULA).
If Q, is a bounded domain with the
Lipschitz boundary and the coefficients aa of a second-order linear differential
operator A are of class
C'a'(Q),
then for arbitrary functions u, v e
Hl'2{Q)
we have
(1.1.4) / (vAu- ulAv)dx = (a0(vdu/dis - udv/dv) + a^uv)dT
where
ao
=
(E(E*;%)2)
'
a*
= - E
a
%
and vk =
YsaJ nj/ao
are components of a so-called conormal to V connected
with the operator A .
Finally, we give a definition of the Fourier transform of a function u e
Lx(Rn):
u(Q = / u(x)exp(-i(x, Q)dx.
1.2. Uniqueness of the continuation. In this section we present up-to-date
results concerning the unique continuation of solutions to a wide class of
partial differential equations. Such a generality requires certain definitions
and notation.
Let m be a multi-index with integer positive components m. satisfying
the following condition: m{ = = mq mq+l We introduce the
^-gradient V^ as the differentiation vector (D{, ... , D , 0, ... , 0).
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